# Criteria for of the existence of uniquely partitionable graphs with respect to additive induced-hereditary properties

• Volume: 22, Issue: 1, page 31-37
• ISSN: 2083-5892

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## Abstract

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Let ₁,₂,...,ₙ be graph properties, a graph G is said to be uniquely (₁,₂, ...,ₙ)-partitionable if there is exactly one (unordered) partition V₁,V₂,...,Vₙ of V(G) such that $G\left[{V}_{i}\right]{\in }_{i}$ for i = 1,2,...,n. We prove that for additive and induced-hereditary properties uniquely (₁,₂,...,ₙ)-partitionable graphs exist if and only if ${}_{i}$ and ${}_{j}$ are either coprime or equal irreducible properties of graphs for every i ≠ j, i,j ∈ 1,2,...,n.

## How to cite

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Izak Broere, Jozef Bucko, and Peter Mihók. "Criteria for of the existence of uniquely partitionable graphs with respect to additive induced-hereditary properties." Discussiones Mathematicae Graph Theory 22.1 (2002): 31-37. <http://eudml.org/doc/270442>.

@article{IzakBroere2002,
abstract = {Let ₁,₂,...,ₙ be graph properties, a graph G is said to be uniquely (₁,₂, ...,ₙ)-partitionable if there is exactly one (unordered) partition V₁,V₂,...,Vₙ of V(G) such that $G[V_i] ∈ _i$ for i = 1,2,...,n. We prove that for additive and induced-hereditary properties uniquely (₁,₂,...,ₙ)-partitionable graphs exist if and only if $_i$ and $_j$ are either coprime or equal irreducible properties of graphs for every i ≠ j, i,j ∈ 1,2,...,n.},
author = {Izak Broere, Jozef Bucko, Peter Mihók},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {induced-hereditary properties; reducibility; divisibility; uniquely partitionable graphs.; partitionable graphs; graph properties; partition; irreducible properties},
language = {eng},
number = {1},
pages = {31-37},
title = {Criteria for of the existence of uniquely partitionable graphs with respect to additive induced-hereditary properties},
url = {http://eudml.org/doc/270442},
volume = {22},
year = {2002},
}

TY - JOUR
AU - Izak Broere
AU - Jozef Bucko
AU - Peter Mihók
TI - Criteria for of the existence of uniquely partitionable graphs with respect to additive induced-hereditary properties
JO - Discussiones Mathematicae Graph Theory
PY - 2002
VL - 22
IS - 1
SP - 31
EP - 37
AB - Let ₁,₂,...,ₙ be graph properties, a graph G is said to be uniquely (₁,₂, ...,ₙ)-partitionable if there is exactly one (unordered) partition V₁,V₂,...,Vₙ of V(G) such that $G[V_i] ∈ _i$ for i = 1,2,...,n. We prove that for additive and induced-hereditary properties uniquely (₁,₂,...,ₙ)-partitionable graphs exist if and only if $_i$ and $_j$ are either coprime or equal irreducible properties of graphs for every i ≠ j, i,j ∈ 1,2,...,n.
LA - eng
KW - induced-hereditary properties; reducibility; divisibility; uniquely partitionable graphs.; partitionable graphs; graph properties; partition; irreducible properties
UR - http://eudml.org/doc/270442
ER -

## References

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2. [3] I. Broere and J. Bucko, Divisibility in additive hereditary graph properties and uniquely partitionable graphs, Tatra Mt. Math. Publ. 18 (1999) 79-87. Zbl0951.05034
3. [4] J. Bucko, M. Frick, P. Mihók and R. Vasky, Uniquely partitionable graphs, Discuss. Math. Graph Theory 17 (1997) 103-114, doi: 10.7151/dmgt.1043. Zbl0906.05057
4. [5] F. Harary, S.T. Hedetniemi and R.W. Robinson, Uniquely colourable graphs, J. Combin. Theory 6 (1969) 264-270, doi: 10.1016/S0021-9800(69)80086-4. Zbl0175.50205
5. [6] P. Mihók, Additive hereditary properties and uniquely partitionable graphs, in: M. Borowiecki and Z. Skupień, eds., Graphs, hypergraphs and matroids (Zielona Góra, 1985) 49-58. Zbl0623.05043
6. [7] P. Mihók, Unique factorization theorem, Discuss. Math. Graph Theory 20 (2000) 143-154, doi: 10.7151/dmgt.1114. Zbl0968.05032

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